Fractal interpolant curve fitting and reproducing kernel Hilbert spaces

In this paper,the linear space F of a special type of fractal interpolation functions (FIFs) on an interval I is considered. Each FIF in F is established from a continuous function on I. We show that, for a finite set of linearly independent continuous functions on I, we get linearly independent FIFs. Then we study a finite-dimensional reproducing kernel Hilbert space (RKHS) FB⊂ F , and the reproducing kernel k for FB is defined by a basis of FB . For a given data set D=\(tₖ, yₖ) : k=0,1,…,N\ , we apply our results to curve fitting problems of minimizing the regularized empirical error based on functions of the form fV+fB , where fV∈ CV and fB∈ FB . Here CV is another finite-dimensional RKHS of some classes of regular continuous functions with the reproducing kernel k^* . We show that the solution function can be written in the form fV+fB=∑ₘ₌₀Nγₘ k^*tₘ +∑ⱼ₌₀N αⱼ ktⱼ , where ktₘ⁽·)= k⁽·,tₘ) and ktⱼ(·)= k(·,tⱼ) , and the coefficients γm and αj can be solved by a system of linear equations.